Calculating Angle Between Plumb Line & Radius Vector Over North Pole

[thesaruman]

Homework Statement

A jet plane flies due south over the north pole with a constant speed of 500 mph. Determine the angle between a plumb line hanging freely in the plane and the radius vector from the center of the Earth to the plane above the north pole.
Hint, assume that the Earth's angular velocity is 2pi radians in 24 hr, which is a good approximation. Why?

Homework Equations

The Attempt at a Solution

Well, I just can't figure out what radius vector is this... I mean, which plane is this? And I didn't understand why the hypothesis of an Earth's angular velocity of 2pi radians in 24hr is an approximation. I thought that this could be considered an exact parameter.

 

[thesaruman]

Homework Statement

A jet plane flies due south over the north pole with a constant speed of 500 mph. Determine the angle between a plumb line hanging freely in the plane and the radius vector from the center of the Earth to the plane above the north pole.
Hint, assume that the Earth's angular velocity is 2pi radians in 24 hr, which is a good approximation. Why?

Homework Equations

dx = v dt is one and v = \omega v dt \sin ( \theta ) is the other.

The Attempt at a Solution

Well, I just can't figure out what radius vector is this... I mean, which plane is this? And I didn't understand why the hypothesis of an Earth's angular velocity of 2pi radians in 24hr is an approximation. I thought that this could be considered an exact parameter.

Considering that this plan is EXACTLY above the North pole, and that in the initial instant of time the jet plane is flying through this radius vector, the answer would depend of a time interval. What I could do?
I mean, the plane would be a distance dx = v dt \hat{\mathbf{x}} in \hat{\mathbf{x}} direction and simultaneously, the Earth would have turned an angle equal to \frac{d\omega}{dt} in \hat{\mathbf{\phi}} direction. I just can't eliminate the time from the solution.

 

[Redbelly98]

First, just cleaning up the LaTex stuff so it displays properly:

... the plane would be a distance
$$dx = v dt \hat{\mathbf{x}} \ \mbox{in} \ \hat{\mathbf{x}} \ \mbox{direction}$$
and simultaneously, the Earth would have turned an angle equal to
$$\frac{d\omega}{dt} \ \mbox{in} \ \hat{\mathbf{\phi}} \ \mbox{direction.}$$

It looks like they want you to find the curve in the plane's path, which is straight line only relative to the rotating Earth. I'm not sure off the top of my head how to do that, but if you can find 3 points along the path it should be possible to fit a circle to them.

I didn't understand why the hypothesis of an Earth's angular velocity of 2pi radians in 24hr is an approximation. I thought that this could be considered an exact parameter.

It has to do with a sidereal vs. solar day.